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2004AugQualEM

# 2004AugQualEM - Ben Sauerwine Practice for Qualifying Exams...

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Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU August 2004 Qualifying Exam Day 2 An electromagnetic wave with fields ( ) 0 0 0 0 ˆ 1 ˆ E z c B e E x E t kz i v v v × = = ω is incident upon a particle of mass m and charge q at the origin. Assume that the motion is non-relatvistic and that the particle is massive so that it stays near the origin. (a) What is the force of the 0 E v field on the particle, and what is the position of the particle as a function of time? ( ) t i t i t i e m qE t x e m qE x x e qE x x m q E F ω ω ω ω = = = = = 2 0 0 0 0 ˆ ˆ & & & & v v (b) Show that an electric dipole ( ) x e E m q t p t i ˆ 0 2 2 ω ω = v is induced. The displacement in Coulomb-meters is then given by ( ) ( ) x e E m q t qx t p t i ˆ 0 2 2 ω ω = = v . (c) For a current density of the form ( ) ( ) t i e x J t x J ω = v v v , the vector potential in the Lorentz gauge can be shown to be given by ( ) ( ) = ' ' ' 4 , ' 3 0 x x e x J x d t x A x x ik v v v v v v v v π μ , with c k ω = .

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2004AugQualEM - Ben Sauerwine Practice for Qualifying Exams...

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